Superadditivity and derivative of operator functions

We will show that if ∑i≠jAiAj≥0 for bounded operators Ai≥0 (i=1,2,⋯,n), then g(∑iAi)≥∑ig(Ai) for every operator convex function g(t) on [0,∞) with g(0)≤0; in particular, (∑iAi)log⁡(∑iAi)≥∑iAilog⁡Ai if each Ai is invertible. Let A,B≥0 and A be invertible. Then we will observe that the Fréchet derivative Dg(sA)(B) is increasing on 0